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Properties

Label 4729.8
Modulus $4729$
Conductor $4729$
Order $788$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4729, base_ring=CyclotomicField(788)) M = H._module chi = DirichletCharacter(H, M([159]))
 
Copy content gp:[g,chi] = znchar(Mod(8, 4729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4729.8");
 

Basic properties

Modulus: \(4729\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4729\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(788\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 4729.l

\(\chi_{4729}(2,\cdot)\) \(\chi_{4729}(3,\cdot)\) \(\chi_{4729}(8,\cdot)\) \(\chi_{4729}(12,\cdot)\) \(\chi_{4729}(18,\cdot)\) \(\chi_{4729}(27,\cdot)\) \(\chi_{4729}(32,\cdot)\) \(\chi_{4729}(48,\cdot)\) \(\chi_{4729}(72,\cdot)\) \(\chi_{4729}(108,\cdot)\) \(\chi_{4729}(121,\cdot)\) \(\chi_{4729}(125,\cdot)\) \(\chi_{4729}(127,\cdot)\) \(\chi_{4729}(128,\cdot)\) \(\chi_{4729}(157,\cdot)\) \(\chi_{4729}(162,\cdot)\) \(\chi_{4729}(175,\cdot)\) \(\chi_{4729}(192,\cdot)\) \(\chi_{4729}(211,\cdot)\) \(\chi_{4729}(227,\cdot)\) \(\chi_{4729}(229,\cdot)\) \(\chi_{4729}(243,\cdot)\) \(\chi_{4729}(245,\cdot)\) \(\chi_{4729}(288,\cdot)\) \(\chi_{4729}(317,\cdot)\) \(\chi_{4729}(335,\cdot)\) \(\chi_{4729}(341,\cdot)\) \(\chi_{4729}(343,\cdot)\) \(\chi_{4729}(367,\cdot)\) \(\chi_{4729}(373,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{788})$
Fixed field: Number field defined by a degree 788 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{159}{788}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(8, a) \) \(1\)\(1\)\(e\left(\frac{65}{394}\right)\)\(e\left(\frac{119}{394}\right)\)\(e\left(\frac{65}{197}\right)\)\(e\left(\frac{261}{394}\right)\)\(e\left(\frac{92}{197}\right)\)\(e\left(\frac{39}{394}\right)\)\(e\left(\frac{195}{394}\right)\)\(e\left(\frac{119}{197}\right)\)\(e\left(\frac{163}{197}\right)\)\(e\left(\frac{429}{788}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 4729 }(8,a) \;\) at \(\;a = \) e.g. 2